We have an answer to "Evaluate the following integrals. Assume a and b..."

Cem Hayes

Answered question

2021-09-25

Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

\int \frac{x}{\sqrt{a x + b}} d x

AGRFTr

Skilled2021-09-26Added 95 answers

Step 1
Consider the integrals,

\int \frac{x}{\sqrt{a x + b}} d x

...(1)
Step 2
Let,

a x + b = t^{2}

To solve,

a x = t^{2} - b

x = \frac{t^{2} - b}{a}

Differentiating w.r.t.t
adx+o=2tdt
adx=2tdt

d x = \frac{2 t}{a} d t

Step 3
Substitute

x = \frac{t^{2} - b}{a} and d x = \frac{2 t}{a} d t

in equation (1)
then,

\int \frac{x}{\sqrt{a x + b}} d x = \int \frac{1}{t} \cdot \frac{t^{2} - b}{a} \cdot \frac{2 t}{a} d t

= \int \frac{2 t (t^{2} - b)}{t \cdot a^{2}} d t

= \frac{2}{a^{2}} \int (t^{2} - b) d t

= \frac{2}{a^{2}} (\frac{t^{3}}{3} - b t) + C

= \frac{2}{a^{2}} [\frac{{\sqrt{a x + b}}^{3}}{3} - b \sqrt{a x + b}] + C

= \frac{2}{a^{2}} [\frac{1}{3} \cdot {(a x + b)}^{\frac{3}{2}} - b \sqrt{a x + b}] + C

Jeffrey Jordon

Expert2021-11-08Added 2605 answers

Answer is given below (on video)

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